The third and fourth videos are not even being played by Totem while for the second one, I can hear the audio but can't see the video. The first one plays well enough. Once again, all these videos work without any problems in 10.04.

I removed totem and made VLC my default. The above solution maybe useful for those who use totem and/or have problems while seeing file properties in nautilus for certain videos (including mp4) as addressed here: askubuntu.com/q/177202/78407
–
ChiragAug 19 '12 at 0:51

you might not be having the codecs in 12.04 which you ahd in 10.04 so first you should try installing the right codec and see
try rhis to install all codecs sudo aptitude install ubuntu-restricted-extras

Instead of removing the particular libgstvideoparsersbad.so, you could uninstall gstreamer0.10-plugins-bad and install gstreamer0.10-plugins-bad-multiverse (from the multiverse repository as the name suggests).

(I had the same problem with totem -> some MP4 files played, others not at all.)

Has nothing to do with the multiverse plugin which only adds encoding support. Better to either patch & rebuild the bad plugin or just remove the currently borked parser plugin as mentioned. The actual decoding is done thru the gstreamer ffmpeg plugin but the parser & other bad plugins do have value &are deps on some apps.
–
dougJul 21 '12 at 6:59